Question

Decide if each of the following statements is true or false. Justify your conclusion with an explanation or counter-example, as appropriate.

1. The zero vector can be part of a basis.
2. Every linearly independent set of vectors is a basis for some subspace.
3. If a vector space V contains a linearly dependent subset of size p, then dimV4. If a vector space V contains a linearly independent subset of size p, then dimV≥p.
5. The dimensions of the row space and column space of a matrix are always equal, even if the matrix is not square.
6. A plane in R³ is a 2-dimensional subspace.
7. If Span(S)=V, then some subest of S is a basis for V.
8. The nonzero rows of A form a basis for Row(A).
9. If B is obtained from A by elementary row operations, then rank(B)=rank(A).
10. If H is a subspace of R³ , then there is a 3×3 matrix A such that A=Col(A).

Answer 1

This detailed answer evaluates each statement regarding vectors, bases, subspaces, and the dimensions of vector spaces in turn, concluding with clear reasons or counterexamples for statements that are false.

Step-by-step explanation:

Let's evaluate the truth of each statement provided:

1. The zero vector cannot be part of a basis because a basis must consist of linearly independent vectors, and including the zero vector would immediately create linear dependence.
2. Every linearly independent set of vectors is indeed a basis for some subspace, specifically the subspace spanned by those vectors.
3. If a vector space V contains a linearly dependent subset of size p, this does not provide information about the dimension, so we cannot determine dimV from this information alone.
4. If a vector space V contains a linearly independent subset of size p, then dimV is at least p because a basis must have at least as many vectors as a linearly independent set.
5. The dimensions of the row space and column space of a matrix are always equal and are known as the rank of the matrix; this holds true regardless of the matrix being square.
6. A plane in R³ is indeed a 2-dimensional subspace because it can be described by two linearly independent vectors.
7. If Span(S)=V, then some subset of S must serve as a basis for V, since the basis is the minimal set of vectors that spans V.
8. The nonzero rows of A do not necessarily form a basis for Row(A) as they need to be linearly independent; however, the row-reduced echelon form of A will have nonzero rows that do form a basis.
9. If B is obtained from A by elementary row operations, then rank(B)=rank(A) because these operations do not change the linear independence of the rows.
10. If H is a subspace of R³, there is not always a 3x3 matrix A such that H=Col(A), because H could be one-dimensional, for example, and not span all of R³.